Point estimates often need to be nested in layers of analysis,and it is the Invariance Principle that provides a pathway fordoing so. An example would be estimating Mu after having toestimate Alpha and Beta for some distributions. In simplerstatistics class exercises (like those we've seen up until now),this is typically avoided by providing the lower level parameterswithin exercises or problems (e.g. asking you for Mu by giving youthe Alpha and Beta). The only real options we've had prior to thisunit for estimating lower level parameters has beentrial-and-error: collecting enough data to form a curve that wethen use probability plots against chosen parameter values until wefind a combination of parameters that "fits" the data we'vecollected. That approach works in the simplest cases, but fails asour problem grows larger and more complex. Even for a singledistribution (e.g., Weibull) there are an infinite number ofpossible Alpha-Beta combinations. We can't manually test themall.
Point estimation gets us around all of that by providing therules needed to actually calculate lower level parameters fromdata. We sometimes need to be able to collect a lot more data touse this approach, but it's worth it. We'll be able to calculatemore than one possible value for many parameters, so it's importantthat we have rules for selecting from among a list ofcandidates.
Discuss what some of those rules are, and how they get appliedin your analysis. If an engineering challenge includes "more thanone reasonable estimator," (Devore, p. 249, Example 6.1 in Section6.1) how do engineers know which to pick, and what issues arisestatistically and in engineering management when making thosechoices?
